Papers by Frances Rosamond
We show that three subclasses of bounded treewidth graphs are well-quasi-ordered by refinements o... more We show that three subclasses of bounded treewidth graphs are well-quasi-ordered by refinements of the minor order. Specifically, we prove that graphs with bounded feedback-vertex-set are well-quasi-ordered by the topological-minor order, graphs with bounded vertex-covers are well-quasiordered by the subgraph order, and graphs with bounded circumference are well-quasi-ordered by the induced-minor order. Our results give an algorithm for recognizing any graph family in these classes which is closed under the corresponding minor order refinement.
Springer eBooks, 2008
In the framework of parameterized complexity, one of the most commonly used structural parameters... more In the framework of parameterized complexity, one of the most commonly used structural parameters is the treewidth of the input graph. The reason for this is that most natural graph problems turn out to be fixed parameter tractable when parameterized by treewidth. However, Graph Layout problems are a notable exception. In particular, no fixed parameter tractable algorithms are known for the Cutwidth, Bandwidth, Imbalance and Distortion problems parameterized by treewidth. In fact, Bandwidth remains NPcomplete even restricted to trees. A possible way to attack graph layout problems is to consider structural parameterizations that are stronger than treewidth. In this paper we study graph layout problems parameterized by the size of the minimum vertex cover of the input graph. We show that all the mentioned problems are fixed parameter tractable. Our basic ingredient is a classical algorithm for Integer Linear Programming when parameterized by dimension, designed by Lenstra and later improved by Kannan. We hope that our results will serve to re-emphasize the importance and utility of this algorithm.
Springer eBooks, Aug 17, 2007
The Convex Recoloring (CR) problem measures how far a tree of characters differs from exhibiting ... more The Convex Recoloring (CR) problem measures how far a tree of characters differs from exhibiting a so-called "perfect phylogeny". For an input consisting of a vertex-colored tree T , the problem is to determine whether recoloring at most k vertices can achieve a convex coloring, meaning by this a coloring where each color class induces a connected subtree. The problem was introduced by Moran and Snir, who showed that CR is NP-hard, and described a search-tree based FPT algorithm with a running time of O(k(k/ log k) k n 4 ). The Moran and Snir result did not provide any nontrivial kernelization. Subsequently, a kernelization with a large polynomial bound was established. Here we give the strongest FPT results to date on this problem: (1) We show that in polynomial time, a problem kernel of size O(k 2 ) can be obtained, and (2) We prove that the problem can be solved in linear time for fixed k. The technique used to establish the second result appears to be of general interest and applicability for bounded treewidth problems.
arXiv (Cornell University), Nov 23, 2018
We are pleased to dedicate this survey on kernelization of the Vertex Cover problem, to Professor... more We are pleased to dedicate this survey on kernelization of the Vertex Cover problem, to Professor Juraj Hromkovič on the occasion of his 60th birthday. The Vertex Cover problem is often referred to as the Drosophila of parameterized complexity. It enjoys a long history. New and worthy perspectives will always be demonstrated first with concrete results here. This survey discusses several research directions in Vertex Cover kernelization. The Barrier Degree of Vertex Cover is discussed. We have reduction rules that kernelize vertices of small degree, including in this paper new results that reduce graphs almost to minimum degree five. Can this process go on forever? What is the minimum vertex-degree barrier for polynomial-time kernelization? Assuming the Exponential-Time Hypothesis, there is a minimum degree barrier. The idea of automated kernelization is discussed. We here report the first experimental results of an AI-guided branching algorithm for Vertex Cover whose logic seems amenable for application in finding reduction rules to kernelize small-degree vertices. The survey highlights a central open problem in parameterized complexity. Happy Birthday, Juraj!
arXiv (Cornell University), Nov 6, 2012
We give an analog of the Myhill-Nerode methods from formal language theory for hypergraphs and us... more We give an analog of the Myhill-Nerode methods from formal language theory for hypergraphs and use it to derive the following results for two NP-hard hypergraph problems. • We provide an algorithm for testing whether a hypergraph has cutwidth at most k that runs in linear time for constant k. In terms of parameterized complexity theory, the problem is fixed-parameter linear parameterized by k. * A preliminary version of this article appeared in the proceedings of ISAAC 2013 [44]. This extended and revised version contains the full proof details, more figures, and corollaries to make the application of the Myhill-Nerode theorem for hypergraphs easier in an algorithmic setting. Moreover, it provides a fix to the proof of the Myhill-Nerode theorem for graphs in the books of Downey and Fellows [14,
Lecture Notes in Computer Science, 2010
The usefulness of parameterized algorithmics has often depended on what Niedermeier has called "t... more The usefulness of parameterized algorithmics has often depended on what Niedermeier has called "the art of problem parameterization". In this paper we introduce and explore a novel but general form of parameterization: the number of numbers. Several classic numerical problems, such as Subset Sum, Partition, 3-Partition, Numerical 3-Dimensional Matching, and Numerical Matching with Target Sums, have multisets of integers as input. We initiate the study of parameterizing these problems by the number of distinct integers in the input. We rely on an FPT result for Integer Linear Programming Feasibility to show that all the abovementioned problems are fixed-parameter tractable when parameterized in this way. In various applied settings, problem inputs often consist in part of multisets of integers or multisets of weighted objects (such as edges in a graph, or jobs to be scheduled). Such number-of-numbers parameterized problems often reduce to subproblems about transition systems of various kinds, parameterized by the size of the system description. We consider several core problems of this kind relevant to number-of-numbers parameterization. Our main hardness result considers the problem: given a non-deterministic Mealy machine M (a finite state automaton outputting a letter on each transition), an input word x, and a census requirement c for the output word specifying how many times each letter of the output alphabet should be written, decide whether there exists a computation of M reading x that outputs a word y that meets the requirement c. We show that this problem is hard for W [1]. If the question is whether there exists an input word x such that a computation of M on x outputs a word that meets c, the problem becomes fixed-parameter tractable.
Algorithmica, Feb 26, 2015
We give an analog of the Myhill-Nerode methods from formal language theory for hypergraphs and us... more We give an analog of the Myhill-Nerode methods from formal language theory for hypergraphs and use it to derive the following results for two NP-hard hypergraph problems. • We provide an algorithm for testing whether a hypergraph has cutwidth at most k that runs in linear time for constant k. In terms of parameterized complexity theory, the problem is fixed-parameter linear parameterized by k. * A preliminary version of this article appeared in the proceedings of ISAAC 2013 [44]. This extended and revised version contains the full proof details, more figures, and corollaries to make the application of the Myhill-Nerode theorem for hypergraphs easier in an algorithmic setting. Moreover, it provides a fix to the proof of the Myhill-Nerode theorem for graphs in the books of Downey and Fellows [14,
Progress in Pattern Recognition, Image Analysis, Computer Vision, and Applications, 2018
This survey is offered in honour of the special occasion of the birthday celebration of science a... more This survey is offered in honour of the special occasion of the birthday celebration of science and education pioneer Professor Juraj Hromkovič. In this survey, we review recent results on one-player floodfilling games on graphs, Flood-It and Free-Flood-It, in which the player aims to make the board monochromatic with a minimum number of flooding moves. As for many colored graph problems, flood-filling games have relevant interpretations in bioinformatics. The original versions of Flood-It and Free-Flood-It are played on n × m grids, but several studies were devoted to analyzing the complexity of these games when the "board" (the graph) belongs to other graph classes. A complete mapping of the complexity of flood-filling games on trees is presented, charting the consequences of single and aggregate parameterizations. The Flood-It problem on trees and the Restricted Shortest Common Supersequence (RSCS) problem are analogous. Flood-It remains NP-hard when played on 3-colored trees. A general framework for reducibility from Flood-It to Free-Flood-It is revisited. The complexity behavior of these games when played on various kinds of graphs is surveyed, such as Cartesian products of cycles and paths, circular grids, split graphs, co-comparability graphs, and AT-free graphs. We review a recent investigation of the parameterized complexity of Flood-It when the size of a minimum vertex cover is the structural parameter. Some educational aspects of the game are also reviewed. Happy Birthday, Juraj!
Lecture Notes in Computer Science, 2012
We motivate and describe a new parameterized approximation paradigm which studies the interaction... more We motivate and describe a new parameterized approximation paradigm which studies the interaction between performance ratio and running time for any parameterization of a given optimization problem. As a key tool, we introduce the concept of α-shrinking transformation, for α ≥ 1. Applying such transformation to a parameterized problem instance decreases the parameter value, while preserving approximation ratio of α (or α-fidelity). For example, it is well-known that Vertex Cover cannot be approximated within any constant factor better than 2 [22] (under usual assumptions). Our parameterized α-approximation algorithm for k-Vertex Cover, parameterized by the solution size, has a running time of 1.273 (2-α)k , where the running time of the best FPT algorithm is 1.273 k [10]. Our algorithms define a continuous tradeoff between running times and approximation ratios, allowing practitioners to appropriately allocate computational resources. Moving even beyond the performance ratio, we call for a new type of approximative kernelization race. Our α-shrinking transformations can be used to obtain kernels which are smaller than the best known for a given problem. For the Vertex Cover problem we obtain a kernel size of 2(2 -α)k. The smaller "αfidelity" kernels allow us to solve exactly problem instances more efficiently, while obtaining an approximate solution for the original instance. We show that such transformations exist for several fundamental problems, including Vertex Cover, d-Hitting Set, Connected Vertex Cover and Steiner Tree. We note that most of our algorithms are easy to implement and are therefore practical in use.
FST TCS 2000: Foundations of Software Technology and Theoretical Computer Science, 2000
We describe some new, simple and apparently general methods for designing FPT algorithms, and ill... more We describe some new, simple and apparently general methods for designing FPT algorithms, and illustrate how these can be used to obtain a significantly improved FPT algorithm for the Maximum Leaf Spanning Tree problem. Furthermore, we sketch how the methods can be applied to a number of other well-known problems, including the parametric dual of Dominating Set (also known as Nonblocker), Matrix Domination, Edge Dominating Set, and Feedback Vertex Set for Undirected Graphs. The main payoffs of these new methods are in improved functions f (k) in the FPT running times, and in general systematic approaches that seem to apply to a wide variety of problems.
Lecture Notes in Computer Science, 2009
We study the Equitable Connected Partition problem, which is the problem of partitioning a graph ... more We study the Equitable Connected Partition problem, which is the problem of partitioning a graph into a given number of partitions, such that each partition induces a connected subgraph, and the partitions differ in size by at most one. We examine the problem from the parameterized complexity perspective with respect to the number of partitions, the treewidth, the pathwidth, the size of a minimum feedback vertex set, the size of a minimum vertex cover, and the maximum number of leaves in a spanning tree of the graph. In particular, we show that the problem is W[1]-hard with respect to the first four parameters (even combined), whereas it becomes fixed-parameter tractable when parameterized by the last two parameters. The hardness result remains true even for planar graphs. We also show that the problem is in XP when parameterized by the treewidth (and hence any other mentioned structural parameter). Furthermore, we show that the closely related problem, Equitable Coloring, is FPT when parameterized by the maximum number of leaves in a spanning tree of the graph.
Lecture Notes in Computer Science, 2008
A graph G on n vertices is a k-leaf power (G ∈ G k ) if it is isomorphic to a graph that can be "... more A graph G on n vertices is a k-leaf power (G ∈ G k ) if it is isomorphic to a graph that can be "generated" from a tree T that has n leaves, by taking the leaves to represent vertices of G, and making two vertices adjacent if and only if they are at distance at most k in T . We address two questions in this paper: (1) As k increases, do we always have Answering an open question of Andreas Brandstädt and Van Bang Le [2, 3, 1], we show that the answer, perhaps surprisingly, is "no." (2) How should one design algorithms to determine, for k-leaf powers, if they have some property? One way this can be done is to use the fact that k-leaf powers have bounded cliquewidth. This fact, plus the FPT cliquewidth approximation algorithm of Oum and Seymour [14], combined with the results of Courcelle, Makowsky and Rotics , allows us to conclude that properties expressible in a general logic formalism, can be decided in FPT time for k-leaf powers, parameterizing by k. This is wildly inefficient. We explore a different approach, under the assumption that a generating tree is given with the graph. We show that one can use the tree directly to decide the property, by means of a finite-state tree automaton. (A more general theorem has been independently obtained by Blumensath and Courcelle [5].) We place our results in a general context of "tree-definable" graph classes, of which k-leaf powers are one particular example.
Lecture Notes in Computer Science
An r-component connected coloring of a graph is a coloring of the vertices so that each color cla... more An r-component connected coloring of a graph is a coloring of the vertices so that each color class induces a subgraph having at most r connected components. The concept has been well-studied for r = 1, in the case of trees, under the rubric of convex coloring, used in modeling perfect phylogenies. Several applications in bioinformatics of connected coloring problems on general graphs are discussed, including analysis of protein-protein interaction networks and protein structure graphs, and of phylogenetic relationships modeled by splits trees. We investigate the r-COMPONENT CONNECTED COLORING COMPLE-TION (r-CCC) problem, that takes as input a partially colored graph, having k uncolored vertices, and asks whether the partial coloring can be completed to an r-component connected coloring. For r = 1 this problem is shown to be NPhard, but fixed-parameter tractable when parameterized by the number of uncolored vertices, solvable in time O * (8 k ). We also show that the 1-CCC problem, parameterized (only) by the treewidth t of the graph, is fixed-parameter tractable; we show this by a method that is of independent interest. The r-CCC problem is shown to be W [1]-hard, when parameterized by the treewidth bound t, for any r ≥ 2. Our proof also shows that the problem is NP-complete for r = 2, for general graphs.
Algorithms and Computation, 2008
In the framework of parameterized complexity, one of the most commonly used structural parameters... more In the framework of parameterized complexity, one of the most commonly used structural parameters is the treewidth of the input graph. The reason for this is that most natural graph problems turn out to be fixed parameter tractable when parameterized by treewidth. However, Graph Layout problems are a notable exception. In particular, no fixed parameter tractable algorithms are known for the Cutwidth, Bandwidth, Imbalance and Distortion problems parameterized by treewidth. In fact, Bandwidth remains NPcomplete even restricted to trees. A possible way to attack graph layout problems is to consider structural parameterizations that are stronger than treewidth. In this paper we study graph layout problems parameterized by the size of the minimum vertex cover of the input graph. We show that all the mentioned problems are fixed parameter tractable. Our basic ingredient is a classical algorithm for Integer Linear Programming when parameterized by dimension, designed by Lenstra and later improved by Kannan. We hope that our results will serve to re-emphasize the importance and utility of this algorithm.
Theory of Computing Systems, 2011
The usefulness of parameterized algorithmics has often depended on what Niedermeier has called "t... more The usefulness of parameterized algorithmics has often depended on what Niedermeier has called "the art of problem parameterization". In this paper we introduce and explore a novel but general form of parameterization: the number of numbers. Several classic numerical problems, such as Subset Sum, Partition, 3-Partition, Numerical 3-Dimensional Matching, and Numerical Matching with Target Sums, have multisets of integers as input. We initiate the study of parameterizing these problems by the number of distinct integers in the input. We rely on an FPT result for Integer Linear Programming Feasibility to show that all the abovementioned problems are fixed-parameter tractable when parameterized in this way. In various applied settings, problem inputs often consist in part of multisets of integers or multisets of weighted objects (such as edges in a graph, or jobs to be scheduled). Such number-of-numbers parameterized problems often reduce to subproblems about transition systems of various kinds, parameterized by the size of the system description. We consider several core problems of this kind relevant to number-of-numbers parameterization. Our main hardness result considers the problem: given a non-deterministic Mealy machine M (a finite state automaton outputting a letter on each transition), an input word x, and a census requirement c for the output word specifying how many times each letter of the output alphabet should be written, decide whether there exists a computation of M reading x that outputs a word y that meets the requirement c. We show that this problem is hard for W [1]. If the question is whether there exists an input word x such that a computation of M on x outputs a word that meets c, the problem becomes fixed-parameter tractable.
Theory of Computing Systems, 2008
The classes of the W-hierarchy are the most important classes of intractable problems in paramete... more The classes of the W-hierarchy are the most important classes of intractable problems in parameterized complexity. These classes were originally defined via the weighted satisfiability problem for Boolean circuits. Here, besides the Boolean connectives we consider connectives such as majority, not-all-equal, and unique. For example, a gate labelled by the majority connective outputs TRUE if more than half of its inputs are TRUE. For any finite set C of connectives we construct the corresponding W(C)-hierarchy. We derive some general conditions which guarantee that the W-hierarchy and the W(C)-hierarchy coincide levelwise. If C only contains the majority connective then the first levels of the hierarchies coincide. We use this to show that a variant of the parameterized vertex cover problem, the majority vertex cover problem, is W[1]-complete.
Electronic Notes in Theoretical Computer Science, 2003
The Graph k-Cut problem is that of finding a set of edges of minimum total weight, in an edge-wei... more The Graph k-Cut problem is that of finding a set of edges of minimum total weight, in an edge-weighted graph, such that their removal from the graph results in a graph having at least k connected components. An algorithm with a running time of O(n k 2 ) for this problem has been known since 1988, due to Goldschmidt and Hochbaum. We show that the problem is hard for the parameterized complexity class W [1]. We also investigate the complexity of a related problem, Cutting A Few Vertices from a Graph, that asks for the minimum cost of separating at least k vertices from an edge-weighted connected graph. We show that this problem also is hard for W [1].
The classes of the W-hierarchy are the most important classes of intractable problems in paramete... more The classes of the W-hierarchy are the most important classes of intractable problems in parameterized complexity. These classes where originally defined via the weighted satisfiability problem for Boolean circuits. Here, besides the Boolean connectives we consider connectives like majority, not-all-equal, and unique. For example, a gate labelled by the majority connective outputs TRUE if more than half of its inputs are TRUE. For any finite set C of connectives we construct the corresponding W(C)-hierarchy. We derive some general conditions which guarantee that the W-hierarchy and the W(C)-hierarchy coincide levelwise. If C only contains the majority connective than the first levels of the hierarchies coincide. We use this to show that a variant of the parameterized vertex cover problem, the majority vertex cover problem, is W[1]-complete.
Milling problem is a natural and quite general graph-theoretic model for geometric milling proble... more Milling problem is a natural and quite general graph-theoretic model for geometric milling problems. Given a graph, one asks for a walk that covers all its vertices with a minimum number of turns, as specified in the graph model by a 0/1 turncost function fx at each vertex x giving, for each ordered pair of edges (e, f ) incident at x, the turn cost at x of a walk that enters the vertex on edge e and departs on edge f . We describe an initial study of the parameterized complexity of the problem. Our main positive result shows that Abstract Milling, parameterized by: number of turns, treewidth and maximum degree, is fixed-parameter tractable, We also show that Abstract Milling parameterized by (only) the number of turns and the pathwidth, is hard for W [1] -one of the few parameterized intractability results for bounded pathwidth.
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Papers by Frances Rosamond